A192737 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)^2/(1 - x*A(x)^3/(1 - x*A(x)^4/(1 - ...))))), a recursive continued fraction.

1, 1, 3, 13, 68, 399, 2531, 16994, 119199, 865849, 6474177, 49616016, 388484212, 3100311228, 25172981053, 207665895001, 1738775327319, 14764815028481, 127076945426555, 1108103873824072, 9787004793441886, 87539719110388691

Offset: 0

Author

Paul D. Hanna, Jul 08 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2531*x^6 +...
The g.f. A = A(x) satisfies A = P(x)/Q(x) where
P(x) = 1 - x*A^2/(1-A) + x^2*A^6/((1-A)*(1-A^2)) - x^3*A^12/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^20/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
Q(x) = 1 - x*A/(1-A) + x^2*A^4/((1-A)*(1-A^2)) - x^3*A^9/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^16/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
Explicitly, the above series begin:
P(x) = exp(1)*(1 - 1/4*x - 283/288*x^2 - 6223/1152*x^3 - 140734037/4147200*x^4 - 3826874463/16588800*x^5 - 290665690804549/175575859200*x^6 +...);
Q(x) = exp(1)*(1 - 5/4*x - 787/288*x^2 - 13731/1152*x^3 - 271921637/4147200*x^4 - 6765586315/16588800*x^5 - 481505562390493/175575859200*x^6 +...).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Crossrefs

Cf. A005169, A192728, A192729, A192730.

Programs

PARI

/* As a recursive continued fraction: */
{a(n)=local(A=1+x, CF); for(i=1, n, CF=1+x; for(k=0, n, CF=1/(1-x*A^(n-k+1)*CF+x*O(x^n))); A=CF); polcoeff(A, n)}

PARI

/* By Ramanujan's continued fraction identity: */
{a(n)=local(A=1+x, P, Q); for(i=1, n,
P=sum(m=0, 2*n, (-x)^m*A^(m*(m+1))/prod(k=1, m, 1-A^k)/exp(1)+x*O(x^(2*n)));
Q=sum(m=0, 2*n, (-x)^m*A^(m^2)/prod(k=1, m, 1-A^k)/exp(1)+x*O(x^(2*n))); A=round(P/Q)); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)) / Product_{k=1..n} (1-A(x)^k),

Q(x) = Sum_{n>=0} (-x)^n * A(x)^(n^2) / Product_{k=1..n} (1-A(x)^k),

due to Ramanujan's continued fraction identity.